2 Equations, formulae and identities

understand that symbols may be used to represent numbers in equations or variables in expressions and formulae

Understand that algebraic expressions follow the generalised rules of arithmetic.

use index notation for positive and negative integer powers, including zero

use index laws in simple cases: xᵐ × xⁿ = xᵐ⁺ⁿ, xᵐ ÷ xⁿ = xᵐ⁻ⁿ and (xᵐ)ⁿ = xᵐⁿ

use index notation involving fractional, negative and zero powers

evaluate expressions by substituting numerical values for letters

collect like terms

multiply a single term over a bracket

take out common factors

expand the product of two simple linear expressions

understand the concept of a quadratic expression and factorise expressions limited to x² + bx + c

expand the product of two or more linear expressions

understand the concept of a quadratic expression and factorise quadratic expressions

manipulate algebraic fractions where the numerator and/or denominator can be numeric, linear or quadratic

complete the square for a given quadratic expression

use algebra to support and construct proofs

understand that a letter may represent an unknown number or a variable

use correct notational conventions for algebraic expressions and formulae

substitute positive and negative integers, decimals and fractions for words and letters in expressions and formulae Evaluate 2x – 3y when x = 4 and y = −5

use formulae from mathematics and other real-life contexts expressed initially in words or diagrammatic form and convert to letters and symbols

derive a formula or expression

change the subject of a formula where the subject appears once

manipulate formulae or equations to change the subject, including cases where the subject appears twice or a power of the subject occurs

solve linear equations with integer or fractional coefficients in one unknown, where the unknown appears on either or both sides

set up simple linear equations from given data The three angles of a triangle are a°, (a + 10)°, (a + 20)°. Find the value of a

set up direct or inverse proportion problems and relate algebraic solutions to graphs, using y ∝ x, y ∝ 1/x, y ∝ x², y ∝ 1/x², y ∝ x³, y ∝ 1/x³, y ∝ √x and y ∝ 1/√x

Calculate exact solutions of two simultaneous linear equations in two unknowns.

Calculate exact solutions of higher-tier simultaneous linear equations in two unknowns.

interpret the equations as lines and the common solution as the point of intersection

solve quadratic equations by factorisation, limited to x² + bx + c = 0

solve quadratic equations by factorisation

solve quadratic equations by using the quadratic formula or completing the square

form and solve quadratic equations from data given in a context

solve simultaneous equations in two unknowns, one linear and one quadratic

2.8.AInequality symbols and compound inequalities

understand and use the symbols >, <, ≥ and ≤, including double-ended inequalities

2.8.BOpen and closed intervals

understand and use the convention for open and closed intervals on a number line

2.8.CSolving linear inequalities

solve simple linear inequalities in one variable and represent the solution set on a number line

2.8.DLinear inequalities on Cartesian graphs

Represent simple linear inequalities on rectangular Cartesian graphs.

2.8.ERegions defined by linear inequalities

identify regions on rectangular Cartesian graphs defined by simple linear inequalities Conventions for the inclusion of boundaries are not required

2.8.HASolving quadratic inequalities

solve quadratic inequalities in one unknown and represent the solution set on a number line

2.8.HBHarder regions defined by linear inequalities

Identify harder regions on Cartesian graphs defined by linear inequalities.